\begin{answer}
    The log likelihood:
    $$
    \begin{aligned}
l(\theta) &= \sum_{i=1}^m\log p(y^{(i)}|x^{(i)}; \theta) \\
&= \sum_{i=1}^m-\log y^{(i)}! + y^{(i)}\theta^Tx^{(i)} - e^{\theta^Tx^{(i)}}
\end{aligned}
$$

And thus 

$$
\begin{aligned}
\frac{\partial l(\theta)}{\partial \theta_j} &= \sum_{i=1}^my^{(i)}x^{(i)}_j - x^{(i)}_j g(\theta^Tx^{(i)})\\
&= \sum_{i=1}^m (y^{(i)} - g(\theta^Tx^{(i)}))x^{(i)}_j
\end{aligned}
$$

So the update rule is given by
$$
\theta_j := \theta_j + \alpha (y^{(i)} - g(\theta^Tx^{(i)}))x_j^{(i)}
$$

    Where $g(\eta) = e^\eta$ is the canonical response function.

\end{answer}
